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Following the 2002 disintegration of the northern and central parts of the Larsen B Ice Shelf, the tributary glaciers of the southern surviving part initially appeared relatively unchanged and hence assumed to be buttressed sufficiently by the remnant ice shelf. Here, we modify this perception with observations from IceBridge altimetry and InSAR-inferred ice flow speeds. Our analyses show that the surfaces of Leppard and Flask glaciers directly upstream from their grounding lines lowered by 15 to 20 m in the period 2002-2011. The thinning appears to be dynamic as the flow of both glaciers and the remnant ice shelf accelerated in the same period. Flask Glacier started accelerating even before the 2002 disintegration, increasing its flow speed by ~55% between 1997 and 2012. Starbuck Glacier meanwhile did not change much. We hypothesize that the different evolutions of the three glaciers are related to their dissimilar bed topographies and degrees of grounding. We apply numerical modeling and data assimilation that show these changes to be accompanied by a reduction in the buttressing afforded by the remnant ice shelf, a weakening of the shear zones between its flow units and an increase in its fracture. The fast flowing northwestern part of the remnant ice shelf exhibits increasing fragmentation, while the stagnant southeastern part seems to be prone to the formation of large rifts, some of which we show have delimited successive calving events. A large rift only 12 km downstream from the grounding line is currently traversing the stagnant part of the ice shelf, defining the likely front of the next large calving event. We propose that the flow acceleration, ice front retreat and enhanced fracture of the remnant Larsen B Ice Shelf presage its approaching demise.